Spinc Manifolds Research Articles

Positive scalar curvature on manifolds with fibered singularities

Abstract A (compact) manifold with fibered 𝑃-singularities is a (possibly) singular pseudomanifold M Σ M_{\Sigma} with two strata: an open nonsingular stratum M ̊ \mathring{M} (a smooth open manifold) and a closed stratum β ⁢ M \beta M (a closed manifold of positive codimension), such that a tubular neighborhood of β ⁢ M \beta M is a fiber bundle with fibers each looking like the cone on a fixed closed manifold 𝑃. We discuss what it means for such an M Σ M_{\Sigma} with fibered 𝑃-singularities to admit an appropriate Riemannian metric of positive scalar curvature, and we give necessary and sufficient conditions (the necessary conditions based on suitable versions of index theory, the sufficient conditions based on surgery methods and homotopy theory) for this to happen when the singularity type 𝑃 is either Z / k \mathbb{Z}/k or S 1 S^{1} , and 𝑀 and the boundary of the tubular neighborhood of the singular stratum are simply connected and carry spin structures. Along the way, we prove some results of perhaps independent interest, concerning metrics on spin𝑐 manifolds with positive “twisted scalar curvature,” where the twisting comes from the curvature of the spin𝑐 line bundle.

Factorization of Dirac Operators on Almost-Regular Fibrations of Spinc Manifolds

Factorization of Dirac Operators on Almost-Regular Fibrations of Spinc Manifolds

Totally umbilical hypersurfaces of Spinc manifolds carrying special spinor fields

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin[Formula: see text] manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin[Formula: see text] case the result of Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to [Formula: see text] is of constant mean curvature. As an application, we prove that there are no extrinsic hypersheres in complete Riemannian [Formula: see text] manifolds of non-constant sectional curvature carrying a parallel, Killing or imaginary Killing spinor.

Uniform K-theory, and Poincaré duality for uniform K-homology

We revisit Špakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology.We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with uniform K-homology and prove Poincaré duality between uniform K-theory and uniform K-homology on spinc manifolds of bounded geometry.

Formal Geometric Quantization III: Functoriality in the Spinc Setting

In this paper, we prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds.

Lower bounds for the eigenvalues of the Spinc Dirac operator on manifolds with boundary

We extend the Friedrich inequality for the eigenvalues of the Dirac operator on Spinc manifolds with boundary under different boundary conditions. The limiting case is then studied and examples are given.

Complex Generalized Killing Spinors on Riemannian Spin c Manifolds

In this paper, we extend the study of generalized Killing spinors on Riemannian Spinc manifolds started by Moroianu and Herzlich to complex Killing functions. We prove that such spinor fields are always real spinc Killing spinors or imaginary generalized Spinc Killing spinors, providing that the dimension of the manifold is greater or equal to 4. Moreover, we examine which Riemannian Spinc manifolds admit imaginary and imaginary generalized Killing spinors.

IIB horizons

We solve the Killing spinor equations for all near-horizon IIB geometries which preserve at least one supersymmetry. We show that generic horizon sections are eight-dimensional almost Hermitian spinc manifolds. Special cases include horizon sections with a Spin(7) structure and those for which the Killing spinor is pure. We also explain how the common sector horizons and the horizons with only 5-form flux are included in our general analysis. We investigate several special cases mainly focusing on the horizons with constant scalars admitting a pure Killing spinor and find that some of these exhibit a generalization of the 2-SCYT condition that arises in the horizons with 5-form fluxes only. We use this to construct new examples of near-horizon geometries with both 3-form and 5-form fluxes.

Geometric K-homology with coefficients II: The Analytic Theory and Isomorphism

AbstractWe discuss the analytic aspects of the geometric model for K-homology with coefficients in ℤ/kℤ constructed in [12]. In particular, using results of Rosenberg and Schochet, we construct a map from this geometric model to its analytic counterpart. Moreover, we show that this map is an isomorphism in the case of a finite CW-complex. The relationship between this map and the Freed-Melrose index theorem is also discussed. Many of these results are analogous to those of Baum and Douglas in the case of spinc manifolds, geometric K-homology, and Atiyah-Singer index theorem.

Seiberg-Witten Equations on Pseudo-Riemannian Spinc Manifolds With Neutral Signature

Abstract Pseudo-Riemannian spinc manifolds were introduced by Ikemakhen in [7]. In the present work we consider pseudo-Riemannian 4-manifolds with neutral signature whose structure groups are SO+(2; 2). We prove that such manifolds have pseudo-Riemannian spinc structure. We construct spinor bundle S and half-spinor bundles S+ and S- on these manifolds. For the first Seiberg-Witten equation we define Dirac operator on these bundles. Due to the neutral metric self-duality of a 2-form is meaningful and it enables us to write down second Seiberg-Witten equation. Lastly we write down the explicit forms of these equations on 4-dimensional at space

Hypersurfaces of Spinc Manifolds and Lawson Type Correspondence

Simply connected three-dimensional homogeneous manifolds $${\mathbb{E}(\kappa, \tau)}$$ , with four-dimensional isometry group, have a canonical Spinc structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into $${\mathbb{E}(\kappa, \tau)}$$ . As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in $${\mathbb{E}(\kappa, \tau)}$$ . Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spinc spinors.

Riemannian manifolds in noncommutative geometry

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spinc manifolds; and conversely, in the presence of a spinc structure. We also show how to obtain an analogue of Kasparov’s fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.

Modular invariance and anomaly cancellation formulas

By studying modular invariance properties of some characteristic forms, we get some new anomaly cancellation formulas. As an application, we derive some results on divisibilities on spin manifolds and congruences on spinc manifolds.

SEIBERG–WITTEN EQUATIONS ON FOUR-DIMENSIONAL LORENTZIAN Spinc MANIFOLDS

New kind of spinors are introduced on four-dimensional Lorentzian Spinc manifolds in [1]. We define Dirac operator on such spinors. In [2] Seiberg–Witten-like equations are written down on Minkowski space. In the present work we generalize these equations to four-dimensional Lorentzian Spinc-manifolds.

THE ENERGY-MOMENTUM TENSOR ON Spinc MANIFOLDS

On Spinc manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a Spinc manifold. Using the notion of generalized cylinders, we derive the variational formula for the Dirac operator under metric deformation and point out that the Energy-Momentum tensor appears naturally as the second fundamental form of an isometric immersion. Finally, we show that generalized Spinc Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors.

The Hijazi Inequalities on Complete Riemannian Spinc Manifolds

We extend the Hijazi type inequality, involving the energy‐momentum tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spinc manifolds without boundary and of finite volume. Under some additional assumptions, using the refined Kato inequality, we prove the Hijazi type inequality for elements of the essential spectrum. The limiting cases are also studied.

Cobordism, Relative Indices and Stein Fillings

In this article we build on the framework developed in Ann. Math. 166, 183–214 ([2007]), 166, 723–777 ([2007]), 167, 1–67 ([2008]) to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the Spin ℂ-Dirac operator with sub-elliptic boundary conditions. We extend our analytic results for sub-elliptic boundary value problems for the Spin ℂ-Dirac operator, and gluing results for the indices of these boundary problems to Spin ℂ-manifolds with several pseudoconvex (pseudoconcave) boundary components. These results are applied to study Stein fillability for compact, 3-dimensional, contact manifolds.

Spin c-manifolds and elliptic genera

We present an extension of the “miraculous cancellation” formulas of Alvarez-Gaumé, Witten and Kefeng Liu to a twisted version where an extra complex line bundle is involved. Relations to the Ochanine congruence formula on 8 k+4 dimensional Spin c manifolds are discussed. To cite this article: F. Han, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Rigidity theorems for SpinC-manifolds

We prove a rigidity and vanishing theorem for Spin C-manifolds. Special cases include the rigidity of the elliptic genus and the elliptic genera of higher level. We state some applications to Spin C-manifolds with nice Pin(2)-action.

Generalized Killing Spinors and Conformal Eigenvalue Estimates for Spin c Manifolds

In this paper we prove the Spinc analog of the Hijazi inequality on the first eigenvalue of the Dirac operator on compact Riemannian manifolds and study its equality case. During this study, we are naturally led to consider generalized Killing spinors on Spinc manifolds and we prove that such objects can only exist on low-dimensional manifolds (up to dimension three). This allows us to give a nice geometrical description of the manifolds satisfying the equality case of the above-mentioned inequality and to classify them in dimension three and four.